Chapter 16. Semilocal Approximations for the Kinetic Energy

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1 Chapter 16 Semilocal Approximations for the Kinetic Energy Fabien Tran 1 and Tomasz A. Wesolowski 2 1 Institute of Materials Chemistry, Vienna University of Technology Getreidemarkt 9/165-TC, A-1060 Vienna, Austria tran@theochem.tuwien.ac.at 2 Department of Physical Chemistry, University of Geneva 30, quai Ernest-Ansermet, CH-1211 Geneva 4, Switzerland tomasz.wesolowski@unige.ch Approximations to the non-interacting kinetic energy T s [ρ], which take the form of semilocal analytic expressions are collected. They are grouped according to the quantities on which they explicitly depend. Additionally, the approximations for quantities related to T s [ρ] kinetic potential and non-additive kinetic energy), for which the analytic expressions for the parent approximation for the functional T s [ρ] are unknown, are also given. Contents 16.1 Notation and conventions Known exact functionals Local density approximation LDA Gradient expansion approximation GEAn Generalized gradient approximation GGA Other semilocal approximations N- andr-dependent approximations Miscellaneous References Notation and conventions Atomic units [m e = e = =1/4πε 0 ) = 1] are used in all formulas. The notation for functions and constants that will be used throughout the text is the following: ρ = ρr) the electron density. s = ρ / 2 3π 2) ) 1/3 ρ 4/3 the dimensionless reduced density gradient. x =5/27) s 2 a quantity proportional to the square of s. C F =3/10) 3π 2) 2/ the Thomas-Fermi constant. 429

2 430 F. Tran & T. A. Wesolowski b =2 6π 2) 1/3 a conversion factor. Capital T integrated kinetic energy. Small t kinetic-energy density per volume unit. The analytic expressions of the non-interacting kinetic-energy functionals T s [ρ] will be given for the spin-compensated case ρ = ρ ). For spin-polarized electron densities, the corresponding expression for T s [ρ,ρ ] can be easily obtained by applying the extension formula of Oliver and Perdew: 1 T s [ρ,ρ ]= 1 2 T s[2ρ ]+T s [2ρ ]) ) The labels used for the approximations reflect the name given by the authors, the most common convention used in the literature, or the names of the authors Known exact functionals For two types of systems, the exact analytic form of T s [ρ] isknown: Thomas and Fermi: 2,3 TF [ρ] =C F ρ 5/3 r)d 3 r ) The Thomas-Fermi functional 2,3 is exact for the homogeneous electron gas. Applying it for inhomogeneous systems leads to an approximation known as the Thomas-Fermi functional or the local density approximation LDA) functional. von Weizsäcker: 4 W [ρ] = 1 2 ρr) d 3 r ) 8 ρr) This functional is exact for one-electron and spin-compensated two-electron systems. Applying it for other systems leads to an approximation known as the von Weizsäcker functional Local density approximation LDA The label LDA is sometimes used in a more general way for any approximation which depends solely on the electron density like the TF functional [Eq )]. In addition to TF, a few such functionals were proposed in the literature. GaussianLDA Lee and Parr: 5 GaussianLDA [ρ] = 3π 2 5/3 ρ 5/3 r)d 3 r )

3 Semilocal Approximations for the Kinetic Energy 431 Note that the coefficient 3π/2 5/ is about 3% larger than the coefficient C F of the TF functional [Eq )]. ZLP Fuentealba and Reyes: 6 ) 1/3 ZLP [ρ] =c 1 ρ 5/3 ρr) r) 1 c 2 ln c 2 ρr) 2 ) 1/3 d 3 r ) where c 1 = and c 2 = It was constructed following the conjointness conjecture 7 applied to the ZLP 8 approximation for the exchange-correlation energy. Note that in Eq. 9) in Ref. 6 the symbol ρ should be replaced by ρ σ for the given coefficients. For the numerical verification see Ref. 9. LP97 Liu and Parr: 10 T LP97 s [ρ] = ρ 5/3 r)d 3 r ) 2 ρ 4/3 r)d 3 r ρ 11/9 r)d 3 r) ) Note that the coefficient in front of the third term was given incorrectly in Ref Gradient expansion approximation GEAn The gradient expansion approximation GEA) until the nth order: n n GEAn [ρ] = T s,i [ρ] = t i ρr), ρr),...) d 3 r ) i=0 i=0 where only the terms for i even are non-zero. The analytical form of the terms up to i = 6 have been derived: t 0 = t TF = C F ρ 5/ ) t 2 = 1 9 tw = 1 ρ ) 72 ρ t 4 = 3π2 ) 2/3 ρ 1/3 2 ) 2 ) ρ 9 2 ρ ρ ρ 8 ρ ρ ρ 4 3 ρ ) ) 3π 2 4/3 t 6 = ρ 13 1/3 2 ρ ) ρ ) 3 ρ ρ 2 4 ρ 144 ρ 16 ρ 2 ρ ρ 2 2 ) 2 ρ 18 ρ ρ 2 ρ 2 ρ ) ρ 36 ρ 2 ρ ) ) 2 ρ ρ 18 ρ ρ ρ 4 72 ρ ρ ρ ρ )

4 432 F. Tran & T. A. Wesolowski These terms were obtained by various authors: t 0 by Thomas and Fermi, 2,3 t 2 by Kompaneets and Pavlovskii 11 and by Kirzhnits, 12 t 4 by Hodges, 13 and t 6 by Murphy. 14 t 2 and t 4 were obtained after integration by part of 15,16 t J 2 = 1 ρ 2 72 ρ t J 4 = 3π2 ) 2/3 ρ 1/ ρ ) 12 4 ρ 2 ρ ) 2 ) 2 ρ 2 ρ 2) 30 ρ ρ ρ ρ ρ 2 ρ 2 2 ρ ρ ρ 3 ρ 2) ρ 3 48 ρ 4 ρ ) respectively. Choosing n = 0, 2, and 4 in Eq ) leads to approximations with the following labels: GEA0 which is just the TF functional [Eq )], GEA2 which is also denoted with TF 1 9W, and GEA Generalized gradient approximation GGA The general form of GGA functionals reads GGA [ρ] = f ρr), ρr) ) d 3 r = C F ρ 5/3 r)f sr))d 3 r ) where F s) is the so-called enhancement factor. The enhancement factor of the von Weizsäcker functional [Eq )] is given by F W s) = 5 3 s ) while for the GEA truncated at the 0th and 2nd orders [Eqs ) )], the enhancement factors are F GEA0 s) = ) F GEA2 s) = s ) respectively. A large group of approximations in the GGA family were constructed following the conjointness conjecture of Lee, Lee, and Parr 7 according to which the enhancement factor of an exchange GGA functional can be used with possible reoptimization of the free coefficients) for the kinetic energy. The following convention is used for labeling the conjoint functionals: if not only the analytic form but also the free coefficients are the same as in the exchange functional, then the functional is called conjoint X, where X stands for the name of the parent exchange functional. In the case of reoptimization of the coefficients, the standard convention applies see Sec. 16.1).

5 Semilocal Approximations for the Kinetic Energy 433 TFλW The functionals in this family differ only in the value of the constant λ: TFλW [ρ] = TF [ρ]+λ W [ρ] ) GEA0 and GEA2 correspond to λ =0and1/9, respectively. Other values of λ were proposed in the literature, e.g., λ =1.290/9 modified 2nd order gradient expansion proposed by Lee et al. 17 ). See Ref. 18 for a compilation of the different values of λ proposed in the literature. The enhancement factor of the TFλW functional is F TFλW s) =1+λ 5 3 s ) P82 Pearson: 19 F P82 s) =1+ 5 s s ) DKPadé DePristo and Kress: 20 F DKPadé x) = x x x x x x x ) LLP Lee, Lee, and Parr: 7 F LLP s) = b 2 s bs arcsinhbs) ) This is the enhancement factor of the exchange functional B88 of Becke 21 refitted for the kinetic energy. OL1 and OL2 Ou-Yang and Levy: 22 F OL1 s) = s π 2 ) 1/3 s ) 3 F OL2 s) = s ) 3π 2 1/3 s C F 1+83π 2 ) 1/3 s ) The coefficients in front of the third terms in OL1 and OL2 were inverted in the original paper of Ou-Yang and Levy caption of Table I in Ref. 22). The correctness of the coefficients given here was confirmed by numerical values of T s [ρ] The correct scaling properties of T s [ρ] require that the coefficient in front of the last terms in OL1 and OL2 are not free but must be non-negative. P92 Perdew: 26 F P92 s) = s s s )

6 434 F. Tran & T. A. Wesolowski T92 Thakkar: 23 F T92 s) = b 2 s bs arcsinhbs) 0.072bs 1+2 5/3 bs ) LC94 Lembarki and Chermette: s arcsinh76.32s) e 100s2) s 2 F LC94 s) = s arcsinh76.32s) s ) This is the enhancement factor of the exchange functional PW91 of Perdew and Wang 28 refitted for the kinetic energy. FR95A Fuentealba and Reyes: 6 F FR95A s) = b 2 s bs arcsinhbs) ) This is the enhancement factor of the exchange functional B88 of Becke 21 refitted for the kinetic energy. FR95B Fuentealba and Reyes: 6 F FR95B s) = s s s 6) 1/ ) This is the enhancement factor of the exchange functional PW86 of Perdew and Wang 29 refitted for the kinetic energy. VSK98 Vitos, Skriver, and Kollár: 30 VJKS00 Vitos et al.: 31 F VSK98 x) = x x x x ) F VJKS00 s) = s s s s ) E00 Ernzerhof: 32 F E00 s) = s2 +5s s ) TW02 Tran and Wesolowski: 25 F TW02 s) =1+κ κ 1+ μ κ s )

7 Semilocal Approximations for the Kinetic Energy 435 where κ = and μ = This is the enhancement factor of the exchange functional PBE of Perdew, Burke, and Ernzerhof 33,34 refitted for the kinetic energy. PBEn Karasiev, Trickey, and Harris: 35 n 1 F PBEn s) =1+ i=1 C n) i s 2 ) i ) 1+a n) s 2 This is the enhancement factor of the exchange functional mpbe of Adamo and Barone 36 refitted for the kinetic energy. Three approximations n = 2, 3, and 4) of the above general form were considered in Ref. 35. The coefficients C n) i and a n) are given in Table Table Coefficients of the enhancement factor of PBEn [Eq )]. Functional C n) 1 C n) 2 C n) 3 a n) PBE PBE PBE exp4 Karasiev, Trickey, and Harris: 35 F exp4 s) =C 1 1 e a 1s 2) + C 2 1 e a 2s 4) ) where C 1 =0.8524, C 2 =1.2264, a 1 = , and a 2 = CR Constantin and Ruzsinszky: 37 F CR s) = 1+ ) a s 2 + a 2 s 4 + a 3 s 6 a 4 s 8 1+a 1 s 2 + a 5 s β 5 a ) 4s 6 Three approximations corresponding to β =1/5, 1/6, and 0.185) of the above general form were considered in Ref. 37. The corresponding sets of coefficients a i are given in Table Table Coefficients of the enhancement factor of CR [Eq )]. β 1/5 1/ a a a a a

8 436 F. Tran & T. A. Wesolowski MGE2 Constantin et al.: 38 F MGE2 κ s) =1+κ 1+ μ ) κ s2 where κ =0.804 and μ = This is the enhancement factor of the exchange functional PBE of Perdew, Burke, and Ernzerhof. 33,34 Conjoint B86A Lacks and Gordon: 39 F B86A b 2 s 2 s)= b 2 s ) This is the enhancement factor of the exchange functional B86A of Becke. 40 Conjoint PW86 Lacks and Gordon: 39 F PW86 s) = s 2 +14s s 6) 1/ ) This is the enhancement factor of the PW86 exchange functional of Perdew and Wang. 29 Conjoint B86B Lacks and Gordon: 39 F B86B b 2 s 2 s)= ) b 2 s 2 ) 4/5 This is the enhancement factor of the exchange functional B86B of Becke. 41 Conjoint DK87 Lacks and Gordon: 39 F DK87 7b 2 s bs s) = π 4 ) 1/ b 2 s ) This is the enhancement factor of the exchange functional DK87 of DePristo and Kress. 42 Conjoint B88 Tran and Wesolowski: 25 F B88 s) = /3 C x b 2 s bs arcsinhbs) ) where C x =3/4) 3/π) 1/3. This is the enhancement factor of the exchange functional B88 of Becke. 21 Conjoint PW91 Lacks and Gordon: 39 F PW91 s) = s arcsinh7.7956s) e 100s2) s s arcsinh7.7956s)+0.004s )

9 Semilocal Approximations for the Kinetic Energy 437 This is the enhancement factor of the exchange functional PW91 of Perdew and Wang. 28 Conjoint xfit Lacks and Gordon: 39 F xfit s) = s s s s s s s 12) ) This is the enhancement factor of the exchange functional xfit of Lacks and Gordon. 43 Conjoint PBE Perdew et al.: 44 F PBE s) =1+κ κ 1+ μ κ s ) where κ =0.804 and μ = This is the enhancement factor of the exchange functional PBE of Perdew, Burke, and Ernzerhof. 33, Other semilocal approximations TB78 Tal and Bader: 45 where TB78 [ρ] = TF [ρ s ]+ W [ρ s ]+ M W [ρ r,a ] ) A=1 ρ r,a r) =ρr A )e 2Z A r R A ) and M ρ s r) =ρr) ρ r,a r) ) M, R A,andZ A are the number, positions, and charges of the nuclei, respectively. A=1 CN84 Cummins and Nordholm: 46 T CN84 s [ρ] = t CN84 r)d 3 r ) where t CN84 r) =max t TF r),t W r) ) )

10 438 F. Tran & T. A. Wesolowski PG85 Pearson and Gordon: 47 T PG85 s [ρ] = t PG85 r)d 3 r ) where { n 1 t PG85 r) = i=0 t 2ir)+ 1 2 t 2nr) if t 2 r) t 0 r) t 0 r) if t 2 r) >t 0 r) ) mgea4 Allan et al.: 48 mgea4 [ρ] = TF [ρ]+ 1 9 T s W [ρ]+ 1 2 T s,4[ρ] ) PP88 Plindov and Pogrebnya: 49 PP88 [ρ] = TF [ρ]+ 1 9 T s W [ρ]+ t 4 r) t 4 r) t 2 r) d 3 r ) MGGA Perdew and Constantin: 50 where and F MGGA = F W + F GE4 M F W) f ab F GE4 M F W) ) F GE4 M = F GEA4 0 z 0 ) b f ab z) = 1+e a/a z) 0 <z<a e a/z +e a/a z) 1 z a 1+ F s2 ) ) ) Equation ) is the enhancement factor of a modified 4th order GEA, where F GEA4 ) and F 4 are the enhancement factors of the functionals GEA4 = t0 + t J 2 + t 4 d 3 r and T s,4 = t 4 d 3 r, respectively [see Eqs ) )]. In Eq ), a = and b = 3 are two parameters which were optimized. GDS08 Ghiringhelli and Delle Site: 51 GDS08 [ρ] = W [ρ]+ ρr)a + B lnρr))) d 3 r ) where A =0.860 and B = MGEA4 Lee et al.: 17 T MGEA4 s [ρ] = t 0 r)+1.789t 2 r) 3.841t 4 r)) d 3 r )

11 Semilocal Approximations for the Kinetic Energy 439 RDA Karasiev et al.: 52 RDA [ρ] = W [ρ]+ t 0 r)f m0 θ r)d 3 r ) where with F m0 θ = A 0 + A 1 κ 4a 1+β 1 κ 4a ) 2 + A 2 κ 4b 1+β 2 κ 4b ) 4 + A 3 κ 2c 1+β 3 κ 2c ) κ 4a = s 4 + ap ) κ 4b = s 4 + bp ) κ 2c = s 2 + cp ) and p = 2 ρ/ 4 3π 2) ) 2/3 ρ 5/3. The constants in Eqs ) ) are A 0 = , A 1 = , A 2 = , A 3 = , β 1 = , β 2 = , β 3 = , a = , b = , and c = N- andr-dependent approximations In the approximations collected below, the analytic expressions for the kinetic energy depend explicitly not only on ρ and its derivatives) but also on the number of electrons N and/or on the position i.e., distance r from the nucleus). The r-dependent expressions are applicable only for mono-atomic systems. HCD84 Haq, Chattaraj, and Deb: 53 HCD84 [ρ] = TF [ρ] 1 r ρr) 40 r 2 d 3 r ) GD94 Ghosh and Deb: 54 GD94 [ρ] = TF [ρ]+ 3 4 ) 1/3 3 π r ρ 4/3 r) 1+ rρ1/3 r) ) d 3 r ) F97 Fuentealba: 55 F97 [ρ] =C F ρ 5/3 1 r) e 8.8ρr)/2)2/3 r d3 r ) 2

12 440 F. Tran & T. A. Wesolowski NLP99 Nagy, Liu, and Parr: 56 T NLP99 s [ρ] = r 2 ρr)d 3 r ) 3 r 2/3 ρr)d 3 r ) 2 r 1 ρr)d 3 r r 1/2 ρr)d 3 r) ) ABSP80 Acharaya et al.: 57 T ABSP80 s [ρ] =T W s [ρ]+ GR82 Gázquez and Robles: 58 T GR82 s [ρ] =T W s [ρ]+ 1 2 N ) ) T TF N 1/3 s [ρ] ) ) T TF N 1/3 N 2/3 s [ρ] ) GB85 Ghosh and Balbás: 59 GB85 [ρ] = W [ρ]+ TF [ρ] ) N 1/3 N 2/3 ) r ρr) r 2 2 ρr) d 3 r ) Miscellaneous Approximation for the kinetic potential model of Chai and Weeks: 60 vt MTF [ρ], r) = 5 3 C Fρ 2/3 r) α 2 ρr) 4 ρr) The modified Thomas-Fermi MTF) ) where α =1/2. Note that there exists no functional MTF such that vt MTF = δ MTF /δρ, because the second term in Eq ) can not be obtained as a functional derivative. Approximation for the non-additive kinetic energy Approximation to the bi-functional nad [ρ A,ρ B ] by Lastra, Kaminski, and Wesolowski: 61 T nadndsd) s [ρ A,ρ B ]=C F ρ A r)+ρ B r)) 5/3 ρ 5/3 A + r) ρ5/3 B r) ) d 3 r f ρ B r), ρ B r))ρ A r) v limit t [ρ B ], r)d 3 r + C[ρ B ] )

13 Semilocal Approximations for the Kinetic Energy 441 where fρ B, ρ B )= vt limit [ρ B ], r) = 1 ρ B r) 2 8 ρ 2 B r) 1 2 ρ B r) 4 ρ B r) ) 1 ) ) 1 e λ s B+s min B ) +1 1 e λ s B+s max B ) ) ) 1 e λ ρ B+ρ min B ) ) with the constants λ = 500, s min B =0.3, s max B =0.9 andρ min B =0.7), and C[ρ B]is a ρ A -independent functional. The acronym NDSD Non-Decomposable approximation to the potential involving only Second Derivatives) reflects the fact that there exists no such functional approximation T s [ρ] from which nadndsd) [ρ A,ρ B ]can be obtained as nadndsd) [ρ A,ρ B ]= T s [ρ A + ρ B ] T s [ρ A ] T s [ρ B ]. References 1. G. L. Oliver and J. P. Perdew, Phys. Rev. A 20, ). 2. L. H. Thomas, Proc. Cambridge Philos. Soc. 23, ). 3. E. Fermi, Z. Phys. 48, ). 4. C. F. von Weizsäcker, Z. Phys. 96, ). 5. C. Lee and R. G. Parr, Phys. Rev. A 35, ). 6. P. Fuentealba and O. Reyes, Chem. Phys. Lett. 232, ). 7. H. Lee, C. T. Lee, and R. G. Parr, Phys. Rev. A 44, ). 8. Q. Zhao, M. Levy, and R. G. Parr, Phys. Rev. A 47, ). 9. T. A. Wesolowski and J. Weber, Int. J. Quantum Chem. 61, ). 10. S. Liu and R. G. Parr, Phys. Rev. A 55, ). 11. A. Kompaneets and E. Pavlovskii, Sov. Phys. JETP 4, ). 12. D. A. Kirzhnits, Sov. Phys. JETP 5, ). 13. C. H. Hodges, Can. J. Phys. 51, ). 14. D. R. Murphy, Phys. Rev. A 24, ). 15. B. K. Jennings, PhD thesis, McMaster University, Canada 1976). 16. M. Brack, B. K. Jennings, and Y. H. Chu, Phys. Lett. B 65, ). 17. D. Lee, L. A. Constantin, J. P. Perdew, and K. Burke, J. Chem. Phys. 130, ). 18. E. V. Ludena and V. V. Karasiev, in Revews of Modern Quantum Chemistry: A Celebration of the Contributions of Robert G Parr, edited by K. Sen World Scientific, Singapore, 2002), pp E. Pearson, PhD thesis, Harvard University, USA 1982). 20. A. E. DePristo and J. D. Kress, Phys. Rev. A 35, ). 21. A. D. Becke, Phys. Rev. A 38, ). 22. H. Ou-Yang and M. Levy, Intl.J.QuantumChem.40, ). 23. A. J. Thakkar, Phys. Rev. A 46, ). 24. Y. A. Bernard, M. Dulak, J. W. Kaminski, and T. A. Wesolowski, J. Phys. A 41, ). 25. F. Tran and T. A. Wesolowski, Int. J. Quantum Chem. 89, ). 26. J. P. Perdew, Phys. Lett. A 165, ). 27. A. Lembarki and H. Chermette, Phys. Rev. A 50, ).

14 442 F. Tran & T. A. Wesolowski 28. J. P. Perdew, Electronic Structure of Solids Akademie Verlag, Berlin, 1991). 29. J. P. Perdew and W. Yue, Phys. Rev. B 33, ). 30. L. Vitos, H. L. Skriver, and J. Kollár, Phys. Rev. B 57, ). 31. L. Vitos, B. Johansson, J. Kollár, and H. L. Skriver, Phys. Rev. A 61, ). 32. M. Ernzerhof, J. Mol. Struct. THEOCHEM) 59, ). 33. J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, ). 34. J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 78, ). 35. V. V. Karasiev, S. B. Trickey, and F. E. Harris, J. Comput.-Aided Mater. Des. 13, ). 36. C. Adamo and V. Barone, J. Chem. Phys. 116, ). 37. L. A. Constantin and A. Ruzsinszky, Phys. Rev. B 79, ). 38. L. A. Constantin, E. Fabiano, S. Laricchia, and F. Della Sala, Phys. Rev. Lett. 106, ). 39. D. J. Lacks and R. G. Gordon, J. Chem. Phys. 100, ). 40. A. D. Becke, J. Chem. Phys. 84, ). 41. A. D. Becke, J. Chem. Phys. 85, ). 42. A. E. DePristo and J. D. Kress, J. Chem. Phys. 86, ). 43. D. J. Lacks and R. G. Gordon, Phys. Rev. A 47, ). 44. J. P. Perdew, M. Ernzerhof, A. Zupan, and K. Burke, J. Chem. Phys. 108, ). 45. Y. Tal and R. Bader, Int. J. Quantum Chem. 12S12), ). 46. P. L. Cummins and S. Nordholm, J. Chem. Phys. 80, ). 47. E. W. Pearson and R. G. Gordon, J. Chem. Phys. 82, ). 48. N. L. Allan, C. G. West, D. L. Cooper, P. J. Grout, and N. H. March, J. Chem. Phys. 83, ). 49. G. Plindov and S. Pogrebnya, Chem. Phys. Lett. 143, ). 50. J. P. Perdew and L. A. Constantin, Phys. Rev. B 75, ). 51. L. M. Ghiringhelli and L. Delle Site, Phys. Rev. B 77, ). 52. V. V. Karasiev, R. S. Jones, S. B. Trickey, and F. E. Harris, Phys. Rev. B 80, ). 53. S. Haq, P. Chattaraj, and B. Deb, Chem. Phys. Lett. 111, ). 54. S. K. Ghosh and B. M. Deb, J. Phys. B 27, ). 55. P. Fuentealba, J. Mol. Struct. THEOCHEM) 390, ). 56. A. Nagy, S. Liu, and R. G. Parr, Phys. Rev. A 59, ). 57. P. K. Acharya, L. J. Bartolotti, S. B. Sears, and R. G. Parr, Proc. Natl. Acad. U.S.A. 77, ). 58. J. L. Gázquez and J. Robles, J. Chem. Phys. 76, ). 59. S. K. Ghosh and L. C. Balbás, J. Chem. Phys. 83, ). 60. J. D. Chai and J. A. Weeks, J. Phys. Chem. B 108, ). 61. J. M. G. Lastra, J. W. Kaminski, and T. A. Wesolowski, J. Chem. Phys. 129, ).